Find the equation of the image of the plane $x-2y+2z-3=0$ in the plane $x+y+z-1=0$.

Question

Find the equation of the image of the plane $x-2y+2z-3=0$ in the plane $x+y+z-1=0$.

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I have no idea how to find the image of a plane in another plane.
Please help me.

Answer 1

It is clear that the planes are not parallel, so they must intersect on a line. If we reflect one plane about the other, obviously it will too pass through this line.

Now the general equation of a plane passing through the intersection of the two planes is $x-2y+2z-3+t(x+y+z-1)=0$. The direction ratios of its normal are $(t+1,t-2,t+2)$.

Since we want the reflection, the new plane and the first should make equal angles with the second plane. Can you find $t$ from here?